Moment-generating function of $2X-Y$

Question

If $X$ and $Y$ are independent random variables, with respective moment-generating functions (mgf) $M_X(t)$ and $M_Y(t)$, the mgf of $2X+Y$ is $M_X(2t)M_Y(t)$. What would the mgf of $2X-Y$ be?

Answer 1

Since $X$ and $Y$ are independent, we have $E(e^{aX}e^{bY})=E(e^{aX})E(e^{bY})$. Thus, we have: $$M_{aX+bY}(t)=E\left(e^{(aX+bY)t}\right)=E\left(e^{(aX)t}e^{(bY)t}\right)=E\left(e^{X(at)}\right)E\left(e^{Y(bt)}\right)=M_{X}(at)M_Y(bt)$$